K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.     

Identities of semigroup algebras of completely O-simple semigroups

Let H = M⁰(G; I, ; P) be a Rees semigroup of matrix type with sandwich matrix P over a group H⁰ with zero. If F is a subgroup of G of finite index and X is a system of representatives of the left cosets of F in G, then with the matrix P there is associated in a natural way a matrix P(F, X) over the group F⁰ with zero. Our main result: the semigroup algebra K[H] of H over a field K of characteristic 0 satisfies an identity if and only if G has an Abelian subgroup F of finite index and, for any X, the matrix P(F, X) has finite determinant rank.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 203–212, August, 1975.     

On some elements of the Brauer group of a conic

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements are linearly independent over ℤ/2ℤ. We construct a field extension K/k and a quaternion algebra D = (u, v) over K such that

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

A relative cohomological invariant for group pairs

We define a cohomological invariantE(G, S, M) whereG is a group,S is a non empty family of (not necessarily distinct) subgroups of infinite index inG andM is a -module ( is the field of two elements). In this paper we are interested in the special case where the family of subgroups consists of just one subgroup, andM is the -module . The invariant will be denoted byE(G, S). We study the relations of this invariant with other endse(G), e(G, S) ande(G,S)), and some results are obtained in the case whereG andS have certain properties of duality.     

On the property <Emphasis Type="Italic">D</Emphasis>(2) and a common splitting field of two biquaternion algebras

Let F be a field of characteristic ≠ 2. We say that F possesses the property D(2) if for any quadratic extension L/F and any two binary quadratic forms over F having a common nonzero value over L, this value can be chosen in F. There exist examples of fields of characteristic 0 that do not satisfy the property D(2). However, as far as we know, it is still unknown whether there are such examples of positive characteristic and what is the minimal 2-cohomological dimension of fields for which the property D(2) does not hold. In this note it is shown that if k is a field of characteristic ≠ 2 such that |k*/k*²| ≥ 4, then for the field k(x) the property D(2) does not hold. Using this fact, we construct two biquaternion algebras over a field K = k(x)((t))((u)) such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e., a field of the kind , where a, b ∈ K*) splitting field. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 242–250.     

Infinite chains of successive normalizers in the general linear group

Let K be a field of characteristics 0 or a field of characteristic 2 and of transcendence degree ≥1, and let G=GL(n, K) be the general linear group of degree n≥2 over K. Further, let . It is proved that in G there exist chains of subgroups {H_m:m ∈ {, infinite in both directions, such that H_m<H_m−1, H_m−1 coincides with the normalizer N_G(H_m), and every quotient group H_m−1/H_m is an elementary Abelian group of type (2,2,...,2) and of rank p. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 30–66. Translated by V. V. Ishkhanov.     

Congruence subgroups of the minimal covolume arithmetic Kleinian group

We consider the problem of finding the normal subgroups of the orientation preserving subgroup Δ⁺ of the [3,5,3]-Coxeter group with the factor group isomorphic to \operatornamePSL₂(\mathbb F_q)\operatorname{\mathrm{PSL}}_{2}(\mathbb {F}_{q}). We identify all such groups with particular congruence subgroups of an arithmetic subgroup of PSL ₂(ℂ) derived from a quaternion algebra over a quartic field. The result can be interpreted as a generalization of the Macbeath’s result on the classification of finite linear groups as Hurwitz groups to 3-dimensional hyperbolic space.     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

On the lattice of subgroups normalized by the symmetric group in a complete monomial group

We consider a lattice of subgroups normalized by the symmetric group S_n in a complete monomial group G = H|S_n, where H is an arbitrary (finite or infinite) group. It is shown that for n3, the subgroup is strongly paranormal in this wreath product for any H. A similar result is obtained for the alternating group A_n, n4. The property of strong paranormality for D in G means that for any element x G, the commutator identity [[x,D],D]=[x, D] holds. This guarantees a standard arrangement of subgroups of G normalized by D. Bibliography: 17 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 111–118.     

Garlands containing the general linear group over an intermediate field

Let K/k be a finite extension of fields with an intermediate subfield L, and let H = GL_L(K) be the general linear group of all L-linear invertible mappings of the vector space of the field K over L. It is proved that the subgroups lying between GL_K(K)^H and the normalizer of H in G, where G = GL_k(K), form a garland. Bibliography: 4 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 34–41.     

The Galois Correspondence in Field Algebra of G-spin Model

Suppose that G is a finite group and D（G） the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D（G; H） of D（G）. This paper gives the concrete construction of a D（G; H）-invariant subspace AH in field algebra of G-spin model and proves that if H is a normal subgroup of G, then AH is Galois closed.     

Lattice of subgroups

Let D be a subgroup of the group G. The lattice of intermediate subgroups is studied. The subgroup F (D F G) is said to be D-complete, if D^F=u:u F>=F. Let F be the subset of all D-complete intermediate subgroups. The system {F, N_G(F)} is a fan for D in G (RZhMat, 1980, 5A208) if and only if D^x> is a D-complete subgroup for any x G. The set {F_ga} coincides with the collection of subgroups of the form D^A (1 A C G) if and only if for any x G the subgroup D, D^x is D-complete. The last condition holds, for example, for a pronormal subgroup D.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 13–19, 1980.     

Weighted group algebras on groups of polynomial growth

Let G be a compactly generated group of polynomial growth and a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L ¹ (G,). In particular, if the weight is sub-exponential, then the algebra L ¹ (G,) is symmetric. For these weights we develop a functional calculus on a total part of L ¹ (G,) and use it to prove the Wiener property. Mathematics Subject Classification (2000):43A20, 22D15, 22D12.Supported by the Austrian Science Foundation project FWF P-14485.Supported by the research grants MEN/CUL/98/007 and CUL/01/014.     

Intertwining operators and polynomials associated with the symmetric group

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators , (i=1, ...,N, where (ij) denotes the transposition of the variablesx _i x _j andk is a fixed parameter). We introduce a family of functions {p }, indexed bym-tuples of non-negative integers = (₁, ..., _m ) formN, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which ,i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatT _i p =0 fori>m, and

The type of group measure space von Neumann algebras

LetG be a countable discrete group acting by measure-preserving automorphisms of a finite measure space (M, ) and let (G,M) be the corresponding group measure space von Neumann algebra, which will be a finite von Neumann algebra. Necessary and sufficient conditions are given for (G,M) to have a non-zero type I part, and the projection on the type I part is explicitly described.This research was supported in part by National Science Foundation Grant MCS 74-19876.